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Ind. Eng. Chem. Fundam. 1985, 2 4 , 500-502
Effect of Mass Transfer on Moving Port Chromatography The effect of mass-transfer resistance and dispersion on chromatographic separations is to cause band spreading and lessening of separation efficiency. Moving port withdrawal of product or injection of feed can improve the efficiency of chromatographic separations, even in the presence of dispersive phenomena.
Introduction The outstanding successes of chromatographic processes on the laboratory bench scale suggest that scale-up to large size separations should be thoroughly explored. Improvements of efficiency and optimization require that accurate mathematical models and methods of solution be available for these processes. The method of moving port chromatography proposed by Wankat (1977,1984) to improve preparative chromatography is one method worthy of further study. Wankat (1977, 1984) utilized a local equilibrium model of chromatography to demonstrate how moving port chromatography can improve separation efficiency. The model omitted mobile-to-stationary-phase mass transfer, intraparticle diffusion, and other rate processes, all of which combine to cause band spreading. In their paper on the moving feed point method, McGary and Wankat (1983) included the effect of longitudinal dispersion. Miller and Wankat (1984), in a computer study of moving port chromatography, estimated the effect of arbitrary degrees of zone spreading that increase as the square root of distance into the column. The purpose of this note is to demonstrate that moment theory can be used to generalize previous treatments to include simply and systematically the dispersive processes. The benefit of moving port chromatography is that it makes efficient use of the separation properties of the packing material. That is, it uses less adsorbent and produces a product that is more concentrated than in normal chromatography. However, aside from the dispersive effects of mass transfer present in all chromatographic processes, moving port chromatography may cause nonuniform concentrations across the column cross section at inlet and outlet ports. Moving port chromatography shares this problem with simulated moving bed chromatography and may require specially designed distributors (Broughton, 1968). To circumvent this difficulty, Wankat (1984) proposed that inlet and outlet ports be located between column sections. If flow from one section to another is through converging and diverging cross-sectional areas, additional zone spreading may be introduced. Wankat (1984) effectively employed the method of characteristics to analyze the linear governing differential equation for the local equilibrium model. Identical information can be obtained by use of moment theory. Each of these two methods has its attributes. The method of characteristics can be applied when an equilibrium relation is nonlinear, e.g., for favorable or unfavorable isotherms. The moment method can be applied to analyze complicated transport processes, provided that the governing partial differential equations are linear (small nonlinearities can be treated by perturbation methods in tandem with moment theory for some systems (Allal et al., 1985)). In the moment method the first moment evaluated at the column exit is the retention time; the second central moment is a measure of the bandwidth and the variance. Included in the column chromatographic processes for which moments are known are fluid-to-solid adsorption (Carbonell and McCoy, 1975),gas-liquid-solid adsorption and partitioning (Alkharasani and McCoy, 1981, 19821,
exclusion effects (Carbonell and McCoy, 1975), and wall-coated open-tube partitioning (Wong et al., 1976). The kinetic and transport processes that have been included are longitudinal dispersion, fluid-to-particle mass transfer, intraparticle diffusion, rate of adsorption, and rate of surface reaction. The moment method has also been utilized for gradient and programming techniques in chromatography (McCoy, 1984, 1979). The temporal moments provide basic information regarding the shapes of chromatographic peaks. The first moment, pl,locates the average position of a chromatographic band and thus provides the same information given by the characteristic lines. The second moment (variance),p2, is a measure of band spreading. For example, for a rectangular input pulse of width to, we have p2 = to2/12. The third central moment, p3, is a measure of asymmetry and, in particular, quantifies the effect of tailing.
Model The moments of chromatographic output pulses are simply related to limits of derivatives of the Laplacetransformed concentration. This fact allows moment expressions to be derived without explicitly solving the coupled partial differential equations describing axial dispersion, film mass transfer, intraparticle diffusion, and linear adsorption processes. The equations for the moments for adsorption chromatography in a column packed with porous spherical particles were first presented by Kubin (1965) and Kucera (1965) and used to analyze experimental data by Schneider and Smith (1968). The expressions for the first absolute and the second central moments are AI1 = (1 + 6 ) z / u (1) with 6 = (1 + p,K/t)t(l - a ) / .
(2)
and with and Y = [~fl/cka
+ (1 + P&/c)'(l/Di
+
5/kpR)R2t/15]c(1- a ) / a ( 5 ) Expressions for third moments are also available (McCoy and Carbonell, 1978). For solute molecules that are too large to penetrate the pores of the particles, one sets t(1- a) = 0, equivalent to taking the volume of pores per unit column volume equal to zero. Since the first moment of a solute is its retention time for the column length z , the solute velocity is obtained as u =
z/pl'
= u/(l
+ 6)
(6)
This is the same result found by the method of characteristics (Wankat, 1984).
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Ind. Eng. Chem. Fundam., Vol. 24, No. 4, 1985 501
f
z
t-
t -
Figure 1. Normal chromatography (no ports along the column) of a mixture of A, B, and C with band spreading shown.
The moment expressions for Apz and Ap3 show the contributions to the change in the moments due to dispersion and mass-transfer resistance. The bandwidth, or square root of the variance, increases as z1I2, according to eq 3. Obviously, the information conveyed by the moments of component concentration profiles is incomplete, even though it is well-defined. Although only the zeroth, first, and second moments are needed to completely define a Gaussian shape, more moments are required for an accurate representation of a more complex concentration profile. Since chromatographic outputs are typically near-Gaussian when D,/vz is small, the moment method provides a simple and useful way to assess quantitatively the dispersive effect to a good first approximation. Discussion For normal chromatography with dispersion, Figure 1 shows how Gaussian bands broaden (in z and t ) as they proceed along the column. In drawing the figures, we have followed McGary and Wankat (1983), who presented graphs for solutes A, B, and C drawn to scale for the system naphthalene, anthracene, and pyrene, respectively, in a-propanol on a poly(vinylpyrro1idone)resin. The solute velocities for these adsorbing species correspond to the ratios U&UB:UC
= 1.15:0.853:0.620
(7)
From eq 6 one finds the ratios
and assuming that axial dispersion is the major masstransfer effect and is the same for the three solutes, one obtains from eq 3 for the second central moments
and for the third moments A ~ ~ A : A ~ ~ &=A0.658:1.611:4.196 /.L~C (10) These results show that solute bands traveling a t lower velocities manifest relatively more spreading and asymmetry (tailing). The reason for this behavior is that solutes with larger residence times within the column are subject to dispersive phenomena for longer times. In Figures 1 and 2 the relative dispersive pulse spreading for components A, B, and C follows eq 9. Dispersive (band spreading) processes cause the characteristic for a trailing edge to be convex and the characteristic for a leading edge to be concave to the horizontal axis. This is consistent with Figure 2 of Miller and Wankat (1984), where amounts of band spreading a t the 1%and 10% levels are shown to vary as z1/2.For a binary mixture, Figure 2 of the present article illustrates the analysis of withdrawal of A and B a t midsection of the column. In this example, A is withdrawn from its leading edge and B from its trailing edge. The plot clearly shows an increase
Figure 2. Withdrawal chromatography (removal through a port at column mid-length) of a mixture of A and B with band spreading shown.
in efficient use of the column. Band broadening due to the usual column mass-transfer phenomena decreases the efficiency of any chromatographic process and does not lessen the effectiveness of moving port chromatography any more than it lessens the effectiveness of normal chromatography. The problem of zone spreading at feed or withdrawal ports is an issue not yet addressed in the literature on moving port chromatography. This question deserves further study, since the essential concept of the process depends upon precise withdrawal or injection with minimum dispersive intrusions. Inadequately timed withdrawals or injections or nonoptimally designed configurations of ports at a column cross section would reduce the separation effectiveness. A perhaps optimistic assumption, consistent with the explanation of Figure 2, is that the band remains Gaussian after withdrawal (or injection), but with reduced (or increased) width as measured by w2lI2. Of course, moand p’ are also altered by the withdrawal or injection. Implicit in Figure 2 is the assumption that the width of band A is reduced by withdrawal a t its leading edge, while the width of band B is reduced by withdrawal at its trailing edge. With similar assumptions a feed port would yield results obvious from earlier discussions, and therefore will not be discussed in any detail here. The straightforward improvements in modeling moving port chromatography presented here, by providing estimates of mass transfer and dispersive behavior, may allow more accurate assessments for scale-up and design of chromatographic separations. Nomenclature D, = effective axial dispersion coefficient, mz/s D i= effective intraparticle diffusion coefficient, m2/s K = equilibrium adsorption constant, m3/ kg k , = adsorption rate coefficient, m3/(s.kg) k p = fluid-to-particle mass-transfer coefficient, m/s to = temporal width of a rectangular pulse of concentration, S
u = solute velocity (eq 6), m/s u = average interstitial velocity, m/s x = defined by eq 4,mz/s y = defined by eq 5, s z = distance coordinate along column, m
Greek Letters a = void fraction c = intraparticle porosity 6 = defined in eq 2 AM = indicates difference between moments at input position ( z = 0) and at z pp = particle density, kg/m3 pl’ = first normalized temporal moment, s p2 = second central, normalized temporal moment, s2 p3 = third central, normalized temporal moment, s3 Literature Cited Alkharasani, M. A,; M c C o y , Aikharasani, M. A,; M c C o y ,
8. J. J . Chromatogr. 1981, 213, 203 B. J. Chem. Eng. J . 1982, 2 3 , 81.
502
Ind. Eng. Chem. Fundam. 1985, 2 4 , 502-504
Allal, K.; Hinds, B. C.; McCoy, B. J. A IChE J ., in press. Broughton, D. B. Chem. Eng. Prog. 1968, 6 4 . 60. Carbonell, R. G.; McCoy, 8. J. Chem. Eng. J . 1975, 9 , 115. Kubin, M. Collect. Czech. Chem. Common. 1985, 3 0 , 1104, 2900 Kucera, K. J . Chromatogr. 1965, 19. 237. McCoy, B. J. S e p . Sci. Techno/. 1979, 1 4 . 515. McCoy, B. J. J . Chromatogr. 1984, 291, 339. McCoy, B. J.; Carbonell, R. G. AIChE J , 1978, 2 4 , 159. McGary, R. S.: Wankat. P. C. Ind. Eng. Chem. Fundam. 1983, 2 2 . 10 Mehta, R . V.; Merson, R. L.; McCoy, B. J. J . Chromatogr. 1974, 88. 1 Miller, G. H.; Wankat, P. C Chem. Eng. Commun. 1984, 3 1 , 21 Schneider. P.; Smith, J. M. A1Gh.E J 1968. 14. 762, 886
Wankat, P. C. I n d . Eng. Chem. Fundam. 1977, 16, 468. Wankat, P. C. Ind. Eng. Chem. Fundam. 1984, 2 3 , 256. Wong. A. K.; McCoy, B. J.; Carbonell, R. G. J . Chromatogr. 1976, 129, 1.
Department of Chemical Engineering 0.niuersity of California Daiis, California 95616
Ben J . McCoy
Received for review September 24, 1984 Accepted July 31, 1985
Use of the Liquid Volume Fraction To Obtain Binary Liquid-Vapor Equilibrium Data without Measuring Composition
A procedure to obtain two-phase densities and compositions from measurements of isochoric pressure, temperature, and liquid volume fractions for binary liquid-vapor systems is presented.
Introduction The equilibrium concentrations and densities of the liquid and vapor phases of a binary system may be determined without direct measurement as shown by simple material balances. The overall density and composition of the mixture in the test cell must be measured or known. Further, if the volumes of liquid and vapor are known, the individual phase compositions and densities can be calculated at a given measured temperature and pressure as shown by Knobler and Scott (1980) and Fontalba et al. (1984). Some of the earliest liquid volume measurements are those by Cragoe and Harper (1921) and Cragoe et al. (1922). Very few measurements of liquid volume fractions were made in the 62 years between 1922 and 1984. But recently, Fontalba et al. (1984) and Merrill et al. (1984) have again undertaken these measurements. Suggested experimental and data reduction procedures based on the measurement of liquid volume fractions are given below. Applications of this method will yield phase concentration and density information. As shown by Fontalba et ai. (1984), a t least two different data points must be obtained for different overall densities at the same temperature and pressure. The following method avoids the need to measure these points directly and allows isochoric data to be used. The theory of liquid volume fractions utilized is an extension of that for pure fluids developed in detail by Van Poolen (1980) and applied lo the prediction of coexistence densities and the critical density in Van Poolen and Haynes (1982) and Van Poolen et al. (1984) Experimental Procedure First, measurements of pressure, temperature, and liquid volume fraction (XLir)are made along isochores for different overall compositions. Two compositions, each represented by the overall concentration of component one (n,and xl’) and two typical isochores ( p T n a t x ,and p T h a t x ,1, are shown in Figure 1.
a parameter at a fixed temperature (Tfix) as illustrated in Figure 2 . The liquid volume fraction can be expressed in terms of the densities from an overall mass balance as given in Van Poolen (1980) as
This is redundant; p’ and p y are constant when T and P are held constant. The original is correct. Also see the parallel treatment in the sentence prior to eq 4. The derivative with respect to overall density, holding pressure and temperature constant, yields
Since p’ and p’ are constant for a fixed pressure and temperature (as described by the Gibbs phase rule), the slope expressed in eq 2 is a constant for all arbitrary points such as “a” and “b” on the line from A to B as shown in Figure 3. (Points a and b are also shown in Figures 1 and 2.) Note that Figure 3 is only used to explain the thermodynamic implications of eq 2 and does not relate directly to data obtained. Equation 2 requires that the points ( P ~ , ~ , X ~ and ”,J from Figures 1 and 2 lie on a straight line on an XLvvs. pT plot. Extending this straight line to XLv= 1 yields p‘, and extending the line to XI,” = 0 obtains p” as illustrated in Figure 4. With the two-phase densities known, the compositions can be found. The mass balance of component one results in another expression for the liquid volume fraction (3)
The derivative with respect to pTxl, holding pressure and temperature constant, yields
(4) Data Analysis a n d Reduction Procedure Graphs of total density and liquid volume fraction vs. pressure are then developed with overall composition as 0196-4313/85/1024-0502$01 50/0
A t a fixed temperature and pressure, the right side of eq 4 is a constant. Therefore, data pairs (pT~l,XI,v), again Cc 1985 American Chemical Society
